reserve n for Nat;

theorem Th37:
  for f be S-Sequence_in_R2 for p be Point of TOP-REAL 2 st p in
  rng f holds L_Cut(f,p) = mid(f,p..f,len f)
proof
  let f be S-Sequence_in_R2;
  let p be Point of TOP-REAL 2;
A1: len f >= 2 by TOPREAL1:def 8;
  assume p in rng f;
  then consider i be Nat such that
A2: i in dom f and
A3: f.i = p by FINSEQ_2:10;
A4: 0+1 <= i by A2,FINSEQ_3:25;
A5: i <= len f by A2,FINSEQ_3:25;
  per cases by A4,XXREAL_0:1;
  suppose
    i > 1;
    then
A6: Index(p,f) + 1 = i by A3,A5,JORDAN3:12;
    then L_Cut(f,p) = mid(f,Index(p,f)+1,len f) by A3,JORDAN3:def 3;
    hence thesis by A2,A3,A6,FINSEQ_5:11;
  end;
  suppose
A7: i = 1;
    thus L_Cut(f,p) = L_Cut(f,f/.i) by A2,A3,PARTFUN1:def 6
      .= f by A7,JORDAN5B:27
      .= mid(f,1,len f) by A1,FINSEQ_6:120,XXREAL_0:2
      .= mid(f,p..f,len f) by A2,A3,A7,FINSEQ_5:11;
  end;
end;
